Distance Between Two Points in a Plane

The Pythagorean theorem can also be used in the coordinate plane – it helps us compute the distance between two points when we know their coordinates.

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Table of contents

• Derivation of the formula
• Formula for the distance
• Solved examples
• Practical uses

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Derivation of the formula

Let us have two points $A=[x_1,y_1]$ and $B=[x_2,y_2]$ in the coordinate plane. The segment $AB$ forms the hypotenuse of a right triangle whose legs are parallel to the coordinate axes.

• The horizontal leg has length $|x_2-x_1|$.
• The vertical leg has length $|y_2-y_1|$.

By the Pythagorean theorem:

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Formula for the distance

> 💡 When squaring a difference, the order doesn't matter – $(x_2-x_1{)}^2=(x_1-x_2{)}^2$. Even negative differences give a positive number after squaring.

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Solved examples

Example 1: Compute the distance between points $A=[1,2]$ and $B=[4,6]$.

Answer: $|AB|=5$.

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Example 2: Compute the distance between points $C=[-2,1]$ and $D=[3,13]$.

Answer: $|CD|=13$.

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Example 3: Compute the distance between points $P=[0,0]$ and $Q=[6,8]$.

Answer: $|PQ|=10$.

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Practical uses

The distance formula is used wherever we work with coordinates:

• 🗺️ Maps and navigation – straight-line distance between two locations on a grid map.
• 🎮 Computer games – computing how far the player is from a target.
• 📐 Geometry – checking whether the sides of a shape are equal in length (for example, whether a triangle is isosceles).
• 📡 Physics – the distance traveled by an object in two-dimensional motion.

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Related articles

• The Pythagorean theorem – complete guide
• The Pythagorean theorem formula
• Word problems
• The Pythagorean theorem in 3D

Practice

• 📊 Word problems
• 🎯 Mixed problems